A local stochastic Lipschitz condition with application to Lasso for high dimensional generalized linear models

TL;DR

This paper establishes exponential tail bounds for the stochastic Lipschitz coefficient in high-dimensional generalized linear models, enhancing understanding of Lasso estimation error bounds in nonlinear settings.

Contribution

It introduces a novel exponential tail inequality for the stochastic Lipschitz coefficient, extending analysis tools from linear to nonlinear high-dimensional models.

Findings

Exponential tail bounds for the Lipschitz coefficient are derived.

Application to Lasso likelihood estimation demonstrates improved error analysis.

Results facilitate better understanding of regularized estimation in nonlinear models.

Abstract

For regularized estimation, the upper tail behavior of the random Lipschitz coefficient associated with empirical loss functions is known to play an important role in the error bound of Lasso for high dimensional generalized linear models. The upper tail behavior is known for linear models but much less so for nonlinear models. We establish exponential type inequalities for the upper tail of the coefficient and illustrate an application of the results to Lasso likelihood estimation for high dimensional generalized linear models.